This paper proposes a novel concept of exponential superiority in probability to compare the numerical methods for general stochastic differential equations from the perspective of the tail probability of the error. We take the linear stochastic oscillator as the test equation and consider several concrete numerical methods. By establishing the large deviation principles of the errors of the considered numerical methods, we show that the symplectic methods are exponentially superior to the non-symplectic methods in probability when the computational time $T$ is sufficiently large. This provides a new way to explain the superiority of stochastic symplectic methods over non-symplectic methods in the long-time simulation.

翻译：本文提出了一种新的概率指数优越性概念，从误差尾部概率的角度比较了一般随机微分方程的数值方法。我们以线性随机振子为测试方程，并考虑了几种具体的数值方法。通过建立各个数值方法的误差的大偏差原理，我们证明了当计算时间 T 足够长时，辛方法在概率上比非辛方法指数优越。这为解释随机辛方法在长时间模拟中的优越性提供了一种新途径。